Question: The numbers 60, 221, and 229 are the legs and hypotenuse of a right triangle.  Find the multiplicative inverse to 450 modulo 3599.  (Express your answer as an integer $n$ with $0\leq n<3599$.)
Solution: We notice that $450=221+229,$ so that must be the connection.  The Pythagorean Theorem tells us  \[60^2+221^2=229^2\] so \[229^2-221^2=60^2.\] The difference of squares factorization tells us  \[(229-221)(229+221)=3600\] and taken modulo 3599 we get  \[8\cdot450\equiv1\pmod{3599}.\] The answer is $\boxed{8}$.